1. In a saucepan, combine chicken breast with chicken stock, one quarter onion, a clove of garlic, and 2 teaspoons salt. Bring to a light boil, and then simmer for 20 minutes. Reserve broth, set chicken aside to cool, and discard onion and garlic. When cool enough to handle, shred chicken with your hands.
2. In a separate saucepan, place tomatillos and serrano chilies in a pot with water, enough to cover them. Bring to boil, and continue boiling until tomatillos turn a different shade of green (from bright green to a dull color). Strain tomatillos and chilies, and place in saucepan with diced onion, minced garlic, and a pinch of salt. Pour in reserved chicken broth, so that liquid just covers the veggies by about an inch. With an immersion (stick) blender, blend all ingredients until they are completely pureed. Bring to a low boil.
3. Pour vegetable oil in a large frying pan over medium/high heat. Slightly fry tortillas one by one in hot oil, setting each on a paper towel afterwards to soak some of the oil. Finally, dip slightly fried tortillas in low-boiling green salsa, until tortillas become soft again. Place on plates, 3 per person.
4. Fill or top tortillas with shredded chicken, then extra green sauce. Top with crumbled cheese, chopped onion, and chopped cilantro.